The Loschmidt Spectral Form Factor

Abstract

The Spectral Form Factor (SFF) measures the fluctuations in the density of states of a Hamiltonian. We consider a generalization of the SFF called the Loschmidt Spectral Form Factor, $\text{tr}[e^{i{H}_{1}T}]\text{tr}[e^{-i{H}_{2}T}]$, for $H_1-H_2$ small. If the ensemble average of the SFF is the variance of the density fluctuations for a single Hamiltonian drawn from the ensemble, the averaged Loschmidt SFF is the covariance for two Hamiltonians drawn from a correlated ensemble. This object is a time-domain version of the parametric correlations studied in the quantum chaos and random matrix literatures. We show analytically that the averaged Loschmidt SFF is proportional to $e^{i\lambda T}T$ for a complex rate $\lambda$ with a positive imaginary part, showing in a quantitative way that the long-time details of the spectrum are exponentially more sensitive to perturbations than the short-time properties. We calculate $\lambda$ in a number of cases, including random matrix theory, theories with a single localized defect, and hydrodynamic theories.

arXiv comment16 pages, comments welcome

References in corpus (29)

• Chaos and Quantum Thermalization
• Thermalization and its mechanism for generic isolated quantum systems
• From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
• Random Matrix Theories in Quantum Physics: Common Concepts
• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Comments on the Sachdev-Ye-Kitaev model
• The Spectrum in the Sachdev-Ye-Kitaev Model
• Dynamics of Loschmidt echoes and fidelity decay
• Lectures on hydrodynamic fluctuations in relativistic theories
• Thermoelectric transport in disordered metals without quasiparticles: the SYK models and holography
• Keldysh technique and non-linear sigma-model: basic principles and applications
• Effective field theory for hydrodynamics: thermodynamics, and the derivative expansion
• Loschmidt Echo
• An introduction to the SYK model
• Spectral statistics and many-body quantum chaos with conserved charge
• Viscosity and dissipative hydrodynamics from effective field theory
• Thouless and relaxation time scales in many-body quantum systems
• Schwinger-Keldysh formalism I: BRST symmetries and superspace
• Effective field theory for dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current
• Dissipation in the effective field theory for hydrodynamics: First order effects
• Spectral statistics in constrained many-body quantum chaotic systems
• Work Statistics, Loschmidt Echo and Information Scrambling in Chaotic Quantum Systems
• Quantum work statistics, Loschmidt echo and information scrambling
• Random Matrix Spectral Form Factor in Kicked Interacting Fermionic Chains
• Spectral Lyapunov exponents in chaotic and localized many-body quantum systems
• Spectral Form Factor of a Quantum Spin Glass
• Spontaneous Symmetry Breaking, Spectral Statistics, and the Ramp
• Fidelity and level correlations in the transition from regularity to chaos
• Parametric Level Correlations in Random-Matrix Models

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